a b s t r a c tWe prove that any linear operator with a kernel in a Gelfand-Shilov space is a composition of two operators with kernels in the same Gelfand-Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten-von Neumann properties for such operators.In this paper we investigate possibilities to decompose linear operators into operators in the same class. It is obvious that for any Banach space B, the set M = B(B) of linear and continuous operators on B is a decomposition algebra. That is, any operator T in M is a composition of two operators in T 1 , T 2 ∈ M, since we may choose T 1 as the identity operator, and T 2 = T . If in addition B is a Hilbert space, then it follows from spectral decomposition that the set of compact operators on B is a decomposition algebra. The decomposition property on compact operators can be obtained by straight-forward application of the spectral theorem. It is therefore more complicated compared to the former one on continuous operators. Note here that the identity operator is not compact when B is not finite-dimensional.An interesting subclass of linear and continuous operators on L 2 concerns the set of all linear operators whose kernels belong to the Schwartz space. There are several proofs of the fact that this operator class is a decomposition algebra (cf. e.g. [1-4] and the references therein).We observe that there are several sets of operators which are not decomposition algebras. For example, if B is an infinitedimensional Hilbert space and 0 < p < ∞, then the set of all Schatten-von Neumann operators of order p is not a decomposition algebra.In this paper we consider the case when M is the set of all linear operators with distribution kernels in the Schwartz class, or in a Gelfand-Shilov space. We note that these operator classes are small, because the restrictions on corresponding kernels are strong. For example, it is obvious that the identity operator does not belong to any of these operator classes.We prove that any such M is a decomposition algebra. Furthermore, in the end of the paper we apply the result and prove that any operator in M as a map between appropriate (quasi-)Banach spaces, belongs to every Schatten-von Neumann class between these spaces.
In this paper, a Fisher information analysis is employed to establish some important physical performance bounds in microwave tomography. As a canonical problem, the two-dimensional electromagnetic inverse problem of imaging a cylinder with isotropic dielectric losses is considered. A fixed resolution is analysed by introducing a finite basis, i.e., pixels representing the material properties. The corresponding Cramér–Rao bound for estimating the pixel values is computed based on a calculation of the sensitivity field which is obtained by differentiating the observed field with respect to the estimated parameter. An optimum trade-off between the accuracy and the resolution is defined based on the Cramér–Rao bound, and its application to assess a practical resolution limit in the inverse problem is discussed. Numerical examples are included to illustrate how the Fisher information analysis can be used to investigate the significance of measurement distance, operating frequency and losses in the canonical tomography set-up.
We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 [20] played a crucial role in rejection of (semi-)classical field models in favor of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favor of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient g (2) (0), is zero (for one photon states), but in (semi-)classical models g (2) (0) ≥ 1. In TSD the coefficient g (2) (0) decreases as 1/E 2 d , where E d > 0 is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient g (2) (0) essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient g (2) (0) on the detection threshold.
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